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%% lecture14.tex
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%% Started on  Thu Jan  5 08:18:04 2012 alex
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\exercises
\begin{xca}
Let us consider a finite cell complex $X$. Its homology
$H_k(X,\ZZ)$ can be represented as a direct sum of free abelian
group $F_k$ and torsion group $T_k$. Prove that its cohomology
$H^k(X,\ZZ)$ is isomorphic to the direct sum of $F_k$ and
$T_{k-1}$.
\end{xca}
\begin{xca}\index{K\"unneth Theorem}
Calculate the homology $H_k(\RP^2\times\RP^2, \ZZ)$ in two ways:
using cell complex and using K\"unneth theorem. (Here $\RP^2$
denotes projective plane.)
\end{xca}
